2.6.2 (1)Please use contradiction to prove. ### 2.6.2(1)Let \( A \subseteq \mathbb{R} \) be a non-empty set. Refer to Example 2.5.13 for the definition of a greatest element of a set; the definition and properties of a least element of a set are completely analogous.(1) Prove that if \( a \in A \) is a greatest element of \( A \), then \( A \) has a least upper bound and \(\text{lub}A = a\). **Definition: Greatest Element**Let \( A \subseteq \mathbb{R} \) be a set, and let \( c \in A \).The number \( c \) is a greatest element of \( A \) if \( x \leq c \) for all \( x \in A \).

Completeness property of R: Every non-empty subsets of real numbers which is bounded above has a…

Answered: Let A CR be a non – empty set. Refer to…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Answer TRUE or FALSE for each of (a)-(e). No justification is required for this problem. (a) {3} ≤ {−1, 1, 3, 5, 6, 9}. (b) {−3,−2, -1,0, 1, 2} = {x ≤ R | − 3 ≤ x ≤ 2}. (c) {6, {6}, (√6)²} ≤ {6, {6}, {{6}}}. (d) If A = {1, 2, 3, 4} and B = {−1,0, 1, 2, 3}, then any relation F from A to B contains exactly four ordered pairs. (e) {y ≤ Z | 0 ≤ y ≤ 5} C {z € R | − 3 < z < 5}.
Decide whether or not the following propositions are true or false. Justify each ofyour conclusions with a proof or a counterexample.(a) ∃x ∈ R (x 2 = x 3 and x > 0)(b) ∀n ∈ N + (2 n + 3 is prime)(c) ¬∃x ∈ R (x 2 < 36)
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6. Use the rules of inference from the formula sheet to show the follow argument is valid: Vx Q(x) 3x (P(x) v Q(x))
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If the contradiction method is used to prove the assertion "For any positive integer n > 0, if a" is even, then a is even.", then a valid implementation of that method begins by assuming that the negation of the statement is true. Choose all of the following statements that correctly express the negation of the above assertion. Vn E N – {0}, rem(a?, 2) = 0 → rem(a, 2) = 0 Vn EN – {0}, rem(a², 2) = 0 rem(a, 2) = 1 En e N – {0}, rem(a?, 2) = 0 → rem(a, 2) In eN - {0}, rem(a², 2) = 0 ^ rem(a, 2) = 1 For any positive integer n > 0, if a" is even, then a is odd. There exists a positive integer n > 0 where a" is even implies that a is od. There exists a positive integer n > 0 where a" is even and a is odd.
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Let A CR be a non – empty set. Refer to Example 2.5.13 for the definition of a greatest element of a set; the definition and properties of a least element of a set are completely analogous. (1)Prove that if a € A is a greatest element of A, then A has a least upper bound and lubA = a.